source file: m1415.txt Date: Thu, 14 May 1998 13:50:57 -0400 Subject: Primes, Lattices, Monzo Planetary Graph From: monz@juno.com (Joseph L Monzo) Paul Erlich wrote: > Joe Monzo wrote, >> Paul Erlich, did you ever consider the fact >> that the relationships between ratios which >> are so elegantly portrayed in your triangular >> lattice diagrams are the result of nothing other >> than prime factorization? > The assertion in that question makes no sense to me, > but Paul Hahn generously replied, >> This doesn't mean all that much--I'm still struggling >> with the 7-limit, but if I were to work in 9-limit or >> higher, I'd probably prefer to diagram the 9s with a >> separate axis, and have one of the unison vectors >> be (2 0 0 -1), to reflect my odd-limit orientation. >> Of course this would be much easier if I were a >> five-dimensional being who could buy 4-dimensional >> graph paper. > I'd have to agree with Paul H., but wonder if there > is something to Joe's question that he wasn't > expressing as well as he would have liked? First comment: I've many times wished that we humans could work with more than 3 dimensions when up against the problem of visualizing and representing more than 3 prime dimensions in music. Fortunately, 7-limit systems _can_ be represented in three dimensions. Response to Paul Erlich: ------------------------ I thought my original question was pretty straightforward, but I never mind elaborating with the help of visual aids. Picking a particular set of pitches arbitrarily, and Going back to Tuning Digest 1315, here's the 7-Limit "Tonality Diamond" in Graham Breed's Triangular Lattice Diagram, which is of the type used by Erlich: 7-Limit Tonality Diamond Lattice, à la Graham Breed -------------------------------------------------- 5/3-------5/4 / \ / \ / \10/7 / \ / 7/6 \ / 7/4 \ / \ / \ 4/3-------1/1-------3/2 \ / \ / \ 8/7 / \12/7 / \ / 7/5 \ / \ / \ / 8/5-------6/5 Graham leaves his 7-factor-ratios unconnected by any lines to the 3- and 5-limit ratios. If I replace the ratios with the prime-factor notation that I use, and connect the 7-ratios in a manner similar to others, I get: Breed Lattice modified by Monzo ------------------------------- Legend: - = 3/2, 4/3 / = 5/4, 8/5 \ = 6/5, 5/3 5^1*3^-1 ---------------- 5^1 / \ / \ / \ 5^1*7^-1 / \ / \ / \ / \ / \ / \ / \ / \/ \ / \ / 7^1*3^-1 -/\-------\/---- 7^1 \ / \ / \ /\ / \ 3^-1 -------\-/---- n^0 --\-/ -------- 3^1 \ / / \ \ / \ 7^-1 -\--/--------\/-3^1*7^-1 / \ \/ /\ / \ /\ / \ / \ / \ / \ / \ / 7^1*5^-1 \ / \ / \ / 5^-1 --------------- 3^1*5^-1 The connecting-lines reveal the major and minor triads, of which there are four each in this tuning. However, the 7-ratios still remain unconnected to the 3- and 5- limit ones. ( Interesting side note inspired by thoughts on ( Schoenberg, who's been discussed here a lot lately: ( in the latter part of his life he strongly reaffirmed ( his faith in Judaism and wrote some pieces on Jewish ( themes, and all his life he apparently had a superstitious ( belief in numerology. If he had seen this diagram, he ( could have been intrigued by the idea of splitting the ( 12-tone row into 2 hexachords, one using the 7-ratio pitches ( connected in the center of the scheme in the form of a Star of ( David to represent a Jewish theme, and the other hexachord ( using the 3- and 5-limit pitches on the perimeter to represent ( a different theme. The difference in prime-limit between ( the two hexachords would harmonically delimit the two sets ( aurally as strongly as they are visually in the diagram. ( Anyone want to collaborate on a piece?) If I take this scale and redraw it onto my lattice diagram, I use a different type of line to connect each prime. same Tonality Diamond, Lattice à la Monzo ----------------------------------------- Here, 6/5 doesn't figure as an interval that needs to be connected directly because it's not a prime axis. Legend: - = 3^1, 3^-1 (= 3/2, 4/3) \ = 5^1, 5^-1 (= 5/4, 8/5) / = 7^1, 7^-1 (= 7/4, 8/7) _ - 5^1 5^1*3^-1 - / _\ - 7^1 \ 3^-1*7^1 -/ \ / \ \ / 5^1*7^-1 \ / \ _ - 3^1 \/ _\ - n^0 - \ / \ 3^-1 - \ / \ 5^-1*7^1 / \ \ / \ _ / - 3^1*7^-1 \ 7^-1 - \/ _ - 3^1*5^-1 5^-1 - I was merely pointing out to Paul that prime factorization figures in his own visual representations of pitch resources. If one is prepared to argue that we don't hear prime qualities in intervals, how can one find a diagram useful which is nothing other than the _visual representation of those qualities_? (I should also note here that all of the diagrams I have seen which were drawn by Erlich himself were representing pitch-classes in 22-Eq temperament and the ratios they _implied_, and not the just ratios themselves. Does this have any bearing on my response here?) Re: Paul Hahn's reply, ---------------------- I think I understand pretty well what you're describing with your > ...diagram the 9s with a separate axis, and have one of > the unison vectors be (2 0 0 -1)... but I'd love to _see_ what this really looks like from your point of view -- try posting a diagram. In my endless search for the "(most) perfect" notation, I've decided in the end to use a variety of representations simultaneously. In addition to the my lattice diagrams, I also use musical staff notation with the prime-factors as accidentals on each note, I also graph the pitches on a 12-eq frequency graph, and I also use what I call a Planetary Graph. This last was my solution to the problem mentioned above of representing more than 3 dimensions. Explanation of my Planetary Graph ================================= Unfortunately, it's impossible for me to draw this within the limited confines of ASCII text. Hopefully, I can get a website up to illustrate it. (My lattice graphs look better with real straight lines too) Anyway, here's a description: 1/1 is represented as the "sun" of the graph, and each increasingly large prime factor is a successive orbit around this sun. The circumference of the orbit represents the "pitch-height" of the ratio as a pitch-class within the octave -- as pitch-height goes higher around the circle, it eventually returns to 12 o'clock at the octave. A line is drawn from the center to the point on the circumference which represents the pitch-height. The prime factorization of the ratio is revealed by little boxes above or below the prime-orbit, each box representing an increase of 1 in the positive and negative exponents respectively. I have found this to be the most compact and efficient way of describing very complex just-intonation systems. At a glance, one can see the relative position of the pitch-classes in frequency as well as how all of them relate by prime factorization. To me, these are the two musically important qualities that ratios possess which I like to see revealed immediately. One of the most interesting things about the Planetary Graph is that it relates back to the ancient Greek and Medieval European ideas about the "Music of the Spheres", the harmonic principles which were felt to underly everything in the universe, from our earthly music to the orbits of the planets and stars. Once again, it is a visual representation I thought I invented, until research for my book showed that the ideas of both representing the octave with a circle and representing ratios with radii had been used around 1600 by Lippius, Descartes, and other theorists of the period. It was useful for showing how ratios and their complements related to each other, and thus for illustrating the then-new concept of inversion in chords. The only new aspect I brought to it was the prime-factor "orbits". Joseph L. Monzo monz@juno.com _____________________________________________________________________ You don't need to buy Internet access to use free Internet e-mail. Get completely free e-mail from Juno at http://www.juno.com Or call Juno at (800) 654-JUNO [654-5866]